A chip-firing game and Dirichlet eigenvalues

Abstract

We consider a variation of the chip-firing game in a induced subgraph S of a graph G. Starting from a given chip configuration, if a vertex v has at least as many chips as its degree, we can fire v by sending one chip along each edge from v to its neighbors. The game continues until no vertex can be fired. We will give an upper bound, in terms of Dirichlet eigenvalues, for the number of firings needed before a game terminates. We also examine the relations among three equinumerous families, the set of spanning forests on S with roots in the boundary of S, a set of "critical" configurations of chips, and a coset group, called the sandpile group associated with S.

Chip-firing games, potential theory on graphs, and spanning trees

“…As we have already mentioned, the key point is that after firing a set A ⊆ V (G)\{q}, the value of b q goes down by exactly the size of A; this makes the function b q a powerful tool for running-time analysis in chip-firing processes. The seemingly different techniques of Tardos [46], Björner, Lovász and Shor [14], Chung and Ellis [19], van den Heuvel [28], and Holroyd, Levine, Mészáros, Peres, Propp and Wilson [29] all give bounds which are specializations of the running-time bound which we derive using b q ; see Remark 5.9.…”

Bijections Between Multiparking Functions, Dirichlet Configurations, and Descending R-Traversals

“…Let DC = DC R, G denote the set of Dirichlet configurations on G with root set R. Dirichlet configurations are usually called critical configurations when G is connected, and this case has been studied extensively (see, for example, [1]). Aspects of Dirichlet configurations were first examined in [4], such as bounds on the number of vertex firings necessary to reach a stable configuration.…”

The chip-firing game

“…Also, there has been an enormous quantity of papers devoted to the general chipfiring game of Björner et al [12] as well as to a generalization to directed graphs [11], see for example [9,20,27,28,31,42]. This generalization to directed graphs has an ancestor called the "probabilistic abacus" in [26].…”